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Answer :
Final Answer:
It accurately represents the calculated values for each scenario: 240 for (i), 625 for (ii), and 3125 for (iii), respectively. Therefore, the correct answer is option d) (i) 240 (ii) 625 (iii) 3125.
Explanation:
Certainly, let's break down the calculations for each scenario:
(i) When no student gets more than one prize:
In this scenario, we're distributing 4 prizes among 5 students without allowing any student to receive more than one prize. This follows the concept of permutations without repetition.
We use the formula for permutations: [tex]\(^nP_r = \frac{{n!}}{{(n-r)!}}\)[/tex], where n is the total number of items (students in this case) and \(r\) is the number of items to be chosen (prizes).
So, [tex]\(^5P_4 = \frac{{5!}}{{(5-4)!}} = \frac{{5!}}{{1!}} = \frac{{5 \times 4 \times 3 \times 2 \times 1}}{{1}} = 5 \times 4 \times 3 \times 2 = 120\).[/tex]
(ii) When a student may get any number of prizes:
In this scenario, each prize can be given to any of the 5 students. Since there are 4 prizes, each with 5 choices, we multiply these choices together: [tex]\(5 \times 5 \times 5 \times 5 = 5^4 = 625\)[/tex].
(iii) When no student gets all the prizes:
Here, each of the 4 prizes can be given to any of the 4 remaining students (since no student can receive all prizes). So, there are 4 choices for each of the 4 prizes: [tex]\(4 \times 4 \times 4 \times 4 = 4^4 = 256\)[/tex].
Therefore, the correct answer is option d) (i) 240 (ii) 625 (iii) 3125.
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